Recent developments of meshfree and particle methods and their applications in applied mechanics are surveyed. Three major methodologies have been reviewed. First, smoothed particle hydrodynamics �SPH � is discussed as a representative of a non-local kernel, strong form collocation approach. Second, mesh-free Galerkin methods, which have been an active research area in recent years, are reviewed. Third, some applications of molecular dynamics �MD � in applied mechanics are discussed. The emphases of this survey are placed on simulations of finite deformations, fracture, strain localization of solids; incompressible as well as compressible flows; and applications of multiscale methods and nano-scale mechanics. This review article includes 397 references. �DOI: 10.1115/1.1431547� 1

. Some of the meshless radial basis function methods used for the numerical solution of partial differential equations are reviewed. In particular, the differences between globally and locally supported methods are discussed, and for locally supported methods the important role of smoothing within a multilevel framework is demonstrated. A possible connection between multigrid finite elements and multilevel radial basis function methods with smoothing is explored. Various numerical examples are also provided throughout the paper. 1. Introduction During the past few years the idea of using so-called meshless methods for the numerical solution of partial differential equations (PDEs) has received much attention throughout the scientific community. As a few representative examples we mention Belytschko and co-worker's results [3] using the so-called element-free Galerkin method, Duarte and Oden's work [11] using h-p clouds, Babuska and Melenk 's work [1] on the partition of unity method, ...

We present a one-parameter family of approximation schemes, which we refer to as local maximumentropy approximation schemes, that bridges continuously two important limits: Delaunay triangulation and maximum-entropy (max-ent) statistical inference. Local max-ent approximation schemes represent a compromise—in the sense of Pareto optimality—between the competing objectives of unbiased statistical inference from the nodal data and the definition of local shape functions of least width. Local max-ent approximation schemes are entirely defined by the node set and the domain of analysis, and the shape functions are positive, interpolate affine functions exactly, and have a weak Kronecker-delta property at the boundary. Local max-ent approximation may be regarded as a regularization, or thermalization, of Delaunay triangulation which effectively resolves the degenerate cases resulting from the lack or uniqueness of the triangulation. Local max-ent approximation schemes can be taken as a convenient basis for the numerical solution of PDEs in the style of meshfree Galerkin methods. In test cases characterized by smooth solutions we find that the accuracy of local max-ent approximation schemes is vastly superior to that of finite elements. Copyright � 2005 John

Reproducing Kernel Particle Methods (RKPM) with a built-in feature of multiresolution analysis are addressed. Some fundamental concepts such as reproducing conditions, and correction function are constructed to systematize the framework of RKPM. In particular, Fourier analysis, as a tool, is exploited to further elaborate RKPM in the frequency domain. Furthermore, we address error estimation and convergence properties. We present several applications which confirm the widespread applicability of multiresolution RKPM. KEY WORDS: reproducing kernel particle methods, multiresolution analysis, wavelets, adaptivity 1 Introduction While traditionally the applications of wavelets were limited to signal and image processing, there has been increasing interest in the utilization of wavelet theory for the solution of partial differential equations (Glowinski, Lawton, Ravachol, and Tenenbaum, 1990; Kurdila, Sun, Grama, and Ko, 1995; Oswald, 1994). Meanwhile, particle Professor of Mechan...

A meshless method is presented which has the advantages of the good meshless methods concerning the ease of introduction of node connectivity in a bounded time of order n, and the condition that the shape functions depend only on the node positions. Furthermore, the method proposed also shares several of the advantages of the Finite Element Method such as: (a) the simplicity of the shape functions in a large part of the domain; (b) C 0 continuity between elements, which allows the treatment of material discontinuities, and (c) ease introduction of the boundary conditions.

The Partition of Unity Method (PUM) can be used to numerically solve a set of differential equations on a domain W. The method is based on the definition of overlapping patches comprising a cover of the domain W. For an efficient implementation it is important that the interaction between the patches themselves, and between the patches and the boundary, is well understood and easily accessible during runtime of the program. We will show that an octree representation of the domain with a tetrahedral mesh at the boundary is an efficient means to provide the needed information. It subdivides an arbitrary domain into simply shaped topological objects (cubes, tetrahedrons) giving a non-overlapping discrete representation of the domain on which efficient numerical integration schemes can be employed. The octants serve as the basic unit to construct the overlapping partitions. The structure of the octree allows the efficient determination of patch interactions.

This work is concerned with developing the hierarchical basis for meshless methods. A reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart. To form such hierarchical partition, a class of basic wavelet functions are introduced. Based upon the built-in consistency conditions, the differential consistency conditions for the hierarchical kernel functions are derived. It serves as an indispensable instrument in establishing the interpolation error estimate, which is theoretically proven and numerically validated. For a special interpolant with different combinations of the hierarchical kernels, a synchronized convergence effect may be observed. Being different from the conventional Legendre function based p- type hierarchical basis, the new hierarchical basis is an intrinsic pseudo-spectral basis, which can remain as a partition of unity in a local region, because the discrete wavelet kernels fo...

The Reproducing Kernel Particle Method (RKPM) has many attractive properties that make it ideal for treating a broad class of physical problems. RKPM may be implemented in a "mesh-full" or a "mesh-free" manner and provides the ability to tune the method, via the selection of a window function and its associated dilation parameter, in order to achieve the requisite numerical performance. RKPM also provides a framework for performing hierarchical computations making it an ideal candidate for simulating multi-scale problems. Although the method has many appealing attributes, it is quite new and its numerical performance is still being quantified with respect to more traditional discretization techniques. In order to assess the numerical performance of RKPM, detailed studies of the method on a series of model partial differential equations has been undertaken. The results of von Neumann analyses for RKPM semi-discretizations of one and two-dimensional, first and second-order ...

In this paper, we modify the robust local image estimation method of R. van den Boomgaard and J. van de Weijer for the approximation of scattered data. The derivation of our knot and data dependent approximation method is based on the relation between the Gaussian facet model in image processing and the moving least square technique known from approximation theory. Numerical examples demonstrate the advantages of our robust scattered data approximation. Key words and phrases: moving least squares, quasi-interpolation, polynomial reproduction, robust estimators, bilateral filters

The implementation of h-adaptivity for meshfree particle methods within a structured framework is described. In this framework, the initial particle arrangement is structured along with a background mesh, and outside boundaries and interior interfaces are described by implicit functions. The advantage of meshfree approximations in this framework lies in the ease of implementing h-adaptivity and the simplicity of the data structures. Particles can easily be added and removed without complications in the data structure, although there are some issues in the quadrature. An a posteriori error estimation is used for the adaptive refinement. An adaptive refinement strategy is applied to several linear elastic problems with high stress and strain gradients and singularities. Several non-linear examples are also given.